{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T14:03:36Z","timestamp":1775829816882,"version":"3.50.1"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"11","license":[{"start":{"date-parts":[[2011,6,9]],"date-time":"2011-06-09T00:00:00Z","timestamp":1307577600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"

\n We define the thin fundamental categorical group\n \n \n \n \n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n M<\/mml:mi>\n ,<\/mml:mo>\n \n \u2217\n \n <\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n {\\mathcal P}_2(M,*)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of a based smooth manifold\n \n \n \n \n (<\/mml:mo>\n M<\/mml:mi>\n ,<\/mml:mo>\n \n \u2217\n \n <\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n (M,*)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n as the categorical group whose objects are rank-1 homotopy classes of based loops on\n \n \n \n M<\/mml:mi>\n M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and whose morphisms are rank-2 homotopy classes of homotopies between based loops on\n \n \n \n M<\/mml:mi>\n M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Here two maps are rank-\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n homotopic, when the rank of the differential of the homotopy between them equals\n \n \n \n n<\/mml:mi>\n n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Let\n \n \n \n \n \n C<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n G<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathcal {C}(\\mathcal {G})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a Lie categorical group coming from a Lie crossed module\n \n \n \n \n \n G<\/mml:mi>\n <\/mml:mrow>\n =<\/mml:mo>\n (<\/mml:mo>\n \n \u2202\n \n <\/mml:mi>\n \n :\n \n <\/mml:mo>\n E<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n G<\/mml:mi>\n ,<\/mml:mo>\n \n \u25b9\n \n <\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n {\\mathcal {G}= (\\partial \\colon E \\to G,\\triangleright )}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . We construct categorical holonomies, defined to be smooth morphisms\n \n \n \n \n \n \n \n P<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n M<\/mml:mi>\n ,<\/mml:mo>\n \n \u2217\n \n <\/mml:mo>\n )<\/mml:mo>\n \n \u2192\n \n <\/mml:mo>\n \n C<\/mml:mi>\n <\/mml:mrow>\n (<\/mml:mo>\n \n G<\/mml:mi>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n {\\mathcal P}_2(M,*) \\to \\mathcal {C}(\\mathcal {G})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , by using a notion of categorical connections, being a pair\n \n \n \n \n (<\/mml:mo>\n \n \u03c9\n \n <\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (\\omega ,m)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n \u03c9\n \n <\/mml:mi>\n \\omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a connection 1-form on\n \n \n \n P<\/mml:mi>\n P<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , a principal\n \n \n \n G<\/mml:mi>\n G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n bundle over\n \n \n \n M<\/mml:mi>\n M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n m<\/mml:mi>\n m<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a 2-form on\n \n \n \n P<\/mml:mi>\n P<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with values in the Lie algebra of\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , with the pair\n \n \n \n \n (<\/mml:mo>\n \n \u03c9\n \n <\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (\\omega ,m)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.\n <\/p>","DOI":"10.1090\/s0002-9947-2010-04857-3","type":"journal-article","created":{"date-parts":[[2010,7,13]],"date-time":"2010-07-13T08:27:49Z","timestamp":1279009669000},"page":"5657-5695","source":"Crossref","is-referenced-by-count":42,"special_numbering":"906","title":["On two-dimensional holonomy"],"prefix":"10.1090","volume":"362","author":[{"given":"Jo\u00e3o","family":"Martins","sequence":"first","affiliation":[]},{"given":"Roger","family":"Picken","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2010,6,9]]},"reference":[{"key":"1","unstructured":"[B] Baez J.C.: Higher Yang-Mills Theory, arXiv:hep-th\/0206130."},{"key":"2","first-page":"492","article-title":"Higher-dimensional algebra. VI. Lie 2-algebras","volume":"12","author":"Baez, John C.","year":"2004","journal-title":"Theory Appl. 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